[FOM] Thoughts on CH 2
Monroe Eskew
meskew at math.uci.edu
Sat Aug 16 20:22:00 EDT 2014
> On Aug 15, 2014, at 11:00 PM, joeshipman at aol.com wrote:
>
> Several times on this forum, I have challenged set theorists to explain what is wrong with the real-valued-measure axiom and I have never gotten a response. From the RVM axiom, all kinds of things can be proven about sets of reals, in fact very little of interest is left undecided by this axiom. It also has the consistency strength of a large cardinal axiom (measurable cardinal). What's not to like?
This virtue of deciding mostly everything about sets of reals is also shared by several other axioms such as CH, MA, PFA. Does RVM do this to a much greater extent?
> I claim that RVM is intuitively consistent even after the Banach-Tarski paradox is taken into account. If mathematics had happened to develop with this assumed as an axiom, then the Banach-Tarski paradox might have been interpreted to mean "space is not isotropic" rather than "space is not infinitely divisible",
If there is any meaningful notion of an isotopic space, then surely Euclidian space is an example of it. It is not the space that fails to be isotopic, but in the case of RVM, the measure. The results of Vitali and later Banach-Tarski establish that a measure that accords with some spatial isotropy cannot exist.
Here's something weird about RVM's. If the continuum is RVM and mu is a total measure on R, then the product measure mu^2 is not a total measure on R^2. Of course a total measure on the plane exists in this situation, but it's not the product measure.
> On some other planet this path might have been taken and their mathematicians might regard our bafflement over CH as a silly prejudice against real-valued measures.
I think something worth asking is why didn't analysts adopt the axiom of RVM at some point. I wouldn't bet the mainstream analysis community feels very constrained by the preferences of set theorists. Perhaps it's not a particularly useful hypothesis. But the same may be said for CH, in terms of the needs of analysts. On the other hand, something like AD in L(R) seems more useful.
Monroe
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